Hierarchical-Absolute Reciprocity Calibration for Millimeter-wave Hybrid Beamforming Systems

The conventional reciprocity calibration (CRC) of HBF was proposed in [27], which is an extension of the relative calibration of the full-digital MIMO system. The CRC treats the HBF system as a virtual full-digital MIMO with $N_{\mathrm{t}}M_{\mathrm{t}}$ virtual antennas and applies OTA signals to estimate the ratio of the transmit and receive mismatch coefficients, which are also called relative calibration coefficients.

In the CRC, the equivalent transmit and receive mismatch coefficients of the BS are defined as $\mathbf{T}_{\mathrm{eq}}=\mathbf{T}_{1}\otimes\mathbf{T}_{2}$ and $\mathbf{R}_{\mathrm{eq}}=\mathbf{R}_{1}\otimes\mathbf{R}_{2}$, and the equivalent mismatch coefficients of the UE are defined as $\mathbf{V}_{\mathrm{eq}}=\mathbf{V}_{1}\otimes\mathbf{V}_{2}$ and $\mathbf{U}_{\mathrm{eq}}=\mathbf{U}_{1}\otimes\mathbf{U}_{2}$. The equivalent uplink and downlink channels are defined as $\mathbf{H}_{\mathrm{UL,eq}}=(\mathbf{r}\otimes\mathbf{R}_{2})\mathbf{H}(\mathbf{v}_{1}\otimes\mathbf{V}_{2})$ and $\mathbf{H}_{\mathrm{DL,eq}}=(\mathbf{u}_{1}\otimes\mathbf{U}_{2})\mathbf{H}^{T}(\mathbf{t}_{1}\otimes\mathbf{T}_{2})$, where $\mathbf{t}_{1}$, $\mathbf{r}_{1}$, $\mathbf{v}_{1}$, and $\mathbf{u}_{1}$ consist of the diagonal entries of $\mathbf{T}_{1}$, $\mathbf{R}_{1}$, $\mathbf{V}_{1}$, and $\mathbf{U}_{1}$, respectively. Based on these definitions, the equation of CRC can be denoted as

where $\mathbf{C}_{\mathrm{BS}}$ and $\mathbf{C}_{\mathrm{UE}}$ represent the relative calibration matrices of the BS and UE.

To obtain the relative calibration coefficients, it is necessary to acquire the equivalent uplink and downlink CSI. To estimate the equivalent downlink CSI, the BS transmits $L_{\mathrm{crc}}$-length pilots by using $Q_{\mathrm{crc}}$ transmit beamforming matrices, and the UE receives the pilots with $P_{\mathrm{crc}}$ receive beamforming matrices, where $P_{\mathrm{crc}}Q_{\mathrm{crc}}=L_{\mathrm{crc}}$. Assume that a $Q_{\mathrm{crc}}$-length pilot sequence is denoted as $\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{Q_{\mathrm{crc}}}\}$, where $\mathbf{x}_{q}$ denotes the pilot during the $[(p-1)Q_{\mathrm{crc}}+q]$-th transmission $(p\in[1:P_{\mathrm{crc}}])$ satisfying $\mathbb{E}\left\{\mathbf{x}_{q}\mathbf{x}_{q}^{H}\right\}=\rho_{\mathrm{c}}\mathbf{I}_{M_{\mathrm{r}}}$. During the training, the digital precoding and combining matrices are set as identity matrices, i.e., $\mathbf{D}_{\mathrm{r}}=\mathbf{I}_{M_{\mathrm{r}}}$ and $\mathbf{W}_{\mathrm{t}}=\mathbf{I}_{M_{\mathrm{t}}}/\sqrt{M_{\mathrm{t}}}$, and the signal received by the UE can be denoted as

where $\mathbf{H}_{\mathrm{DL}}=\mathbf{U}_{2}\mathbf{H}^{T}\mathbf{T}_{2}$, $\mathbf{F}_{\mathrm{t},q}$ denotes the analog beamforming matrix at the BS, $\mathbf{B}_{\mathrm{r},q}$ is the analog combining matrix at the UE, $\mathbf{F}_{\mathrm{eq},q}=\mathrm{blkdiag}(\mathbf{f}_{q,1},\mathbf{f}_{q,2},\cdots,\mathbf{f}_{q,M_{\mathrm{t}}})$, $\mathbf{f}_{q,m}$ denotes the $m$-th column of $\mathbf{F}_{\mathrm{t},q}$, $\mathbf{B}_{\mathrm{eq},p}=\mathrm{blkdiag}(\mathbf{b}_{p,1},\mathbf{b}_{p,2},\cdots,\mathbf{b}_{p,M_{\mathrm{r}}})$, and $\mathbf{b}_{p,m}$ is the $m$-th row of $\mathbf{B}_{\mathrm{r},p}$.

By stacking all $L_{\mathrm{crc}}$-length signals in matrix form denoted as $\mathbf{Y}_{\mathrm{UE}}=[\bar{\mathbf{Y}}_{\mathrm{UE},1}^{T},\cdots,\bar{\mathbf{Y}}_{\mathrm{UE},P_{\mathrm{crc}}}^{T}]^{T}\in\mathbb{C}^{M_{\mathrm{r}}P_{\mathrm{crc}}\times Q_{\mathrm{crc}}}$ and $\bar{\mathbf{Y}}_{\mathrm{UE},p}=[\mathbf{y}_{\mathrm{UE},p,1},\cdots,\mathbf{y}_{\mathrm{UE},p,Q_{\mathrm{crc}}}]$, the received signal model can be given by

where $\tilde{\mathbf{B}}_{\mathrm{eq}}=[\mathbf{B}_{\mathrm{eq},1}^{T},\mathbf{B}_{\mathrm{eq},2}^{T},\cdots,\mathbf{B}_{\mathrm{eq},P_{\mathrm{crc}}}^{T}]^{T}\in\mathbb{C}^{M_{\mathrm{r}}P_{\mathrm{crc}}\times N_{\mathrm{r}}}$, $\tilde{\mathbf{F}}_{\mathrm{eq}}=[\mathbf{F}_{\mathrm{eq},1}\mathbf{x}_{1},\mathbf{F}_{\mathrm{eq},2}\mathbf{x}_{2},\cdots,\mathbf{F}_{\mathrm{eq},Q_{\mathrm{crc}}}\mathbf{x}_{Q_{\mathrm{crc}}}]\in\mathbb{C}^{N_{\mathrm{t}}\times Q_{\mathrm{crc}}}$, $\mathbf{N}_{\mathrm{eq}}=[\bar{\mathbf{N}}_{\mathrm{eq},1}^{T},\cdots,\bar{\mathbf{N}}_{\mathrm{eq},P_{\mathrm{crc}}}^{T}]^{T}$, and $\bar{\mathbf{N}}_{\mathrm{eq},p}=[\mathbf{n}_{\mathrm{eq},p,1},\cdots,\mathbf{n}_{\mathrm{eq},p,Q_{\mathrm{crc}}}]$. By vectoring the matrix $\mathbf{Y}_{\mathrm{UE}}$, the received signal can be further denoted as

where $\mathbf{B}_{\mathrm{crc}}=(\tilde{\mathbf{F}}_{\mathrm{eq}}^{T}\otimes\tilde{\mathbf{B}}_{\mathrm{eq}})$. Using the LS approach[36], the equivalent downlink channel is estimated as

Similarly, to estimate the equivalent uplink channel, the UE transmit the uplink training pilots to the BS. Further, to estimate the calibration coefficients, the UE feeds back the estimated downlink channel to the BS.

After the BS estimates the uplink channel and receives the downlink channel fed back from the UE, the calibration coefficients can be computed by the following proposition.

To measure the complexity of the CRC, we further derive the overhead and computational complexity. The overhead of reciprocity calibration can be expressed by the count of channel use for transmitting calibration signals and feeding back the estimated CSI. The computational complexity can be measured by the times of multiplication for estimating the channel state information and computing the mismatch coefficients.

In the CRC, the dimensions of the equivalent channels $\mathbf{H}_{\mathrm{UL,eq}}$ and $\mathbf{H}_{\mathrm{DL,eq}}$ (see (6)) are much larger than that of the actual wireless channel matrix $\mathbf{H}$ (see (1)), which generates the heavy overhead of the channel estimation and high computational complexity. Further, since the CRC only estimates the ratio of the mismatch coefficients of transmit chains and receive chains, mmWave channel estimation of mmWave systems, which requires the knowledge of the individual mismatch coefficients, becomes invalid. Thus, due to limitations of the CRC in fully connected mmWave-HBF systems, we propose a hierarchical-absolute calibration (HAC) approach, which decouples the reciprocity calibration of digital RF chains and analog RF chains and estimates the individual mismatch coefficients of transmit chains and receive chains, respectively.

To reduce the overhead and complexity of the reciprocity calibration of fully connected HBF systems, digital RF chains and analog RF chains must be calibrated individually, which means hierarchical calibration. To adopt the mmWave channel estimation approaches of mmWave systems, the individual mismatch coefficients are required rather than the ratio of the mismatch coefficients, which can be addressed by applying the absolute reciprocity calibration.

However, due to the fully-connected structure of the HBF system, HAC encounters two challenges. On the one hand, the digital RF chains and analog RF chains are physically coupled via the fully-connected phase shift network, which results in the decoupling challenge. On the other hand, the fully-connected phase shift network causes that the RF chains can not transmit and receive signals independently, which results in the calibration challenge. To a certain degree, these problems can be addressed by using extra auxiliary circuits to assistant the reciprocity calibration, e.g. the calibration approaches presented in [37, 38]. But the auxiliary circuits may bring extra non-reciprocity, and the calibration accuracy of hardware-circuit calibration highly depends on the auxiliary circuits [20]. Thus, we propose an OTA-based HAC for mmWave-HBF systems. Specifically, the calibrations of digital and analog RF chains are decoupled by a targeted beamforming scheme, and the mismatch coefficients are estimated by the OTA training signals between the BS and UE.

In the rest of this section, we will introduce the decoupling principle of the proposed HAC. Since the transmitter and receiver employ similar HBF structures, the decouple operation is first explained in the multi-input single-output (MISO) system for clarity, and then we will propose the concrete design for the general HBF-MIMO system.

MISO system for decoupling digital chains from the analog chains: We consider a MISO system where the transmitter is equipped with the fully-connected HBF structure and the receiver is equipped with a single antenna as illustrated in Fig. 2a. Thanks to the fully-connected structure, the signal transmitted from each digital RF chain passes through all antennas in the transmitter and all wireless channels. For example, if the $m$-th transmit digital chain transmit a signal $x_{m}$ to the receiver, the received signal can be denoted as

where $h_{\mathrm{eq},m}$ denotes the virtual equivalent channel between the $m$-th transmit digital chain and the receive antenna. Based on this, the HBF MISO system can be virtually constructed as a DBF MISO system as illustrated in Fig. 2b, where the virtual antennas are the transmit digital chains, and the virtual-equivalent channels consist of the phase shift network, the analog RF chains, and the wireless channel. Further, it can be found that the virtual-equivalent channels equal to each other when the beamforming vectors are identical, i.e., $\mathbf{f}_{1}=\cdots=\mathbf{f}_{M}$. Thus, by applying this analog beamforming design, the digital chains can be decoupled from the analog chains, and the absolute reciprocity calibration can be considered as the calibration with known channel gains.

MISO system for decoupling the analog chains from the digital chains: Since each digital RF chain is connected to all analog RF chains via the fully-connected phase shifter network, the antenna array can be considered as an ABF system as shown in Fig. 3a. By using only one digital RF chain to transmit and receive calibration signals, the analog RF chains can be decoupled from the digital RF chains as illustrated in Fig. 3b. Based on this design, the analog RF chains can be calibrated with signal processing approaches.

Concrete design: Based on the above MISO systems, the decoupling principle can be extended to a general case where both the BS and UE are equipped with multiple antennas and HBF structures as illustrated in Fig. 1. For general point-to-point HBF MIMO systems, the overall HAC training phases can be divided into two phases, which are downlink training and uplink training as illustrated in Fig. 4. During the training processes, the calibration training signals and beamforming matrices of the BS and UE should be designed to decouple digital and analog RF chains. By taking the downlink training phase as an example, the concrete designs are given as follows.

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  Downlink training pilots: The entire $L_{\mathrm{d}}$-length downlink training pilots consist of $L_{\mathrm{dr}}$-length pilots for calibrating digital RF chains and $L_{\mathrm{da}}$-length pilots for calibrating the analog RF chains, where $L_{\mathrm{dr}}+L_{\mathrm{da}}=L_{\mathrm{d}}$. To increase the degree of freedom of received signals, the $L_{\mathrm{dr}}$-length pilots are transmitted by using $Q_{\mathrm{dr}}$ transmit beamforming matrices and received by using $P_{\mathrm{dr}}$ receive beamforming matrices, where $L_{\mathrm{dr}}=Q_{\mathrm{dr}}P_{\mathrm{dr}}$. The $L_{\mathrm{da}}$-length pilots possess the homologous structure, i.e., $L_{\mathrm{da}}=Q_{\mathrm{da}}P_{\mathrm{da}}$. By using $\{\mathbf{x}_{1},\cdots,\mathbf{x}_{Q_{\mathrm{max}}}\}$ to represent the pilots set, the transmitted pilot $\mathbf{x}_{\mathrm{d},l}$ during the $l$-th transmission can be denoted as $\mathbf{x}_{\mathrm{d},l}=\mathbf{x}_{q}$, where $\mathbb{E}\left\{\mathbf{x}_{q}\mathbf{x}_{q}^{H}\right\}=\rho_{\mathrm{c}}\mathbf{I}_{\mathrm{M}_{t}}$, $q=l\%Q_{\mathrm{dr}}$ when $l\leq L_{\mathrm{dr}}$, and $q=(l-L_{\mathrm{dr}})\%Q_{\mathrm{da}}$ when $L_{\mathrm{da}}<l\leq L_{\mathrm{d}}$, $Q_{\mathrm{max}}=\max\{Q_{\mathrm{dr}},Q_{\mathrm{da}}\}$.

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  Beamforming design for calibrating digital chains: Let $\mathbf{F}_{\mathrm{dr},q}\in\mathbb{C}^{N_{\mathrm{t}}\times M_{\mathrm{t}}}$ denote the analog transmit beamforming matrix and $\mathbf{B}_{\mathrm{dr},p}\in\mathbb{C}^{N_{\mathrm{r}}\times M_{\mathrm{r}}}$ represent the receive beamforming matrix during the $l$-th transmission, where $l\leq L_{\mathrm{dr}}$, $q=l\%Q_{\mathrm{dr}}$, and $p=l\%P_{\mathbf{dr}}$. To decouple the mismatch of the digital chains from the analog chains, analog beamforming matrices are designed as $\mathbf{F}_{\mathrm{dr},1}=\cdots=\mathbf{F}_{\mathrm{dr},Q_{\mathrm{dr}}}=\mathbf{f}_{\mathrm{dr}}\mathbf{1}_{M_{\mathrm{t}}}^{T}$ and $\mathbf{B}_{\mathrm{dr},1}=\cdots=\mathbf{B}_{\mathrm{dr},P_{\mathrm{dr}}}=\mathbf{b}_{\mathrm{dr}}\mathbf{1}_{M_{\mathrm{r}}}^{T}$, where each element of $\mathbf{f}_{\mathrm{dr}}\in\mathbb{C}^{N_{\mathrm{t}}}$ and $\mathbf{b}_{\mathrm{dr}}\in\mathbb{C}^{N_{\mathrm{r}}}$ possesses random phase. Further, the digital precoding matrix can be designed as $\mathbf{W}_{\mathrm{dr},q}=\mathbf{I}_{M_{\mathrm{t}}}/\sqrt{M_{\mathrm{t}}}$ and the digital receive combining matrix is given by $\mathbf{D}_{\mathrm{dr},p}=\mathbf{I}_{M_{\mathrm{r}}}$ during the downlink training phase.

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  Beamforming design for calibrating analog chains: During the $l$-th transmission ($l>L_{\mathrm{dr}}$), the analog transmit beamforming matrix $\mathbf{F}_{\mathrm{da},q}$ and receive beamforming matrix $\mathbf{B}_{\mathrm{da},p}$ can be designed as random phase matrices, i.e., the elements of $\mathbf{F}_{\mathrm{da},q}$ and $\mathbf{B}_{\mathrm{da},p}$ possess random phases. Let $\mathbf{W}_{\mathrm{da},q}$ denote the digital precoding matrix and $\mathbf{D}_{\mathrm{da},p}$ denote the digital receive combining matrix, where $q=(l-L_{\mathrm{dr}})\%Q_{\mathrm{da}}$, and $p=(l-L_{\mathrm{dr}})\%P_{\mathrm{da}}$. To decouple the mismatch of analog RF chains from the mismatch of digital RF chains, the digital precoding matrix can be designed as $\mathbf{W}_{\mathrm{da},q}=\mathrm{blkdiag}(1,\mathbf{0}_{M_{\mathrm{t}}-1,M_{\mathrm{t}}-1})$, and the digital receive combining matrix can be given by $\mathbf{D}_{\mathrm{da},p}=\mathrm{blkdiag}(1,\mathbf{0}_{M_{\mathrm{r}}-1,M_{\mathrm{r}}-1})$.

Since the mismatch coefficients of transmit chains have nothing to do with that of receive chains, HAC can be divided into downlink HAC and uplink HAC. The downlink HAC is applied to calibrate the transmit chains of the BS and the receive chains of the UE, while the uplink HAC can calibrate the receive chains of the BS as well as the transmit chains of the UE. The uplink HAC is similar to the downlink HAC. Thus, we introduce the signal modeling, the problem formulation, and the problem decoupling by taking the downlink HAC as an example.

By considering the BS transmits $l$-th pilot to the UE, the signal received by the UE can be modeled as

where $\tilde{\mathbf{n}}_{\mathrm{d},l}=\mathbf{D}_{\mathrm{r},l}^{T}\mathbf{U}_{1}\mathbf{B}_{\mathrm{r},l}^{T}\mathbf{n}_{\mathrm{d},l}$. When $l\leq L_{\mathrm{dr}}$, $\mathbf{D}_{\mathrm{r},l}=\mathbf{D}_{\mathrm{dr},p}$, $\mathbf{B}_{\mathrm{r},l}=\mathbf{B}_{\mathrm{dr},p}$, $\mathbf{F}_{\mathrm{t},q}=\mathbf{F}_{\mathrm{dr},q}$, and $\mathbf{W}_{\mathrm{t},l}=\mathbf{W}_{\mathrm{dr},q}$, where $p=l\%P_{\mathrm{dr}}$, and $q=l\%Q_{\mathrm{dr}}$. When $l>L_{\mathrm{dr}}$, $\mathbf{D}_{\mathrm{r},l}=\mathbf{D}_{\mathrm{da},p}$, $\mathbf{B}_{\mathrm{r},l}=\mathbf{B}_{\mathrm{da},p}$, $\mathbf{F}_{\mathrm{t},q}=\mathbf{F}_{\mathrm{da},q}$, and $\mathbf{W}_{\mathrm{t},l}=\mathbf{W}_{\mathrm{da},q}$, where $p=(l-L_{\mathrm{dr}})\%P_{\mathrm{da}}$, and $q=(l-L_{\mathrm{dr}})\%Q_{\mathrm{da}}$.

After the BS transmits $L_{\mathrm{d}}$-length pilots to the UE, the optimization problem for jointly estimating $\mathbf{U}_{1},\ \mathbf{U}_{2},\ \mathbf{T}_{1},\ \mathbf{T}_{2}$, and $\mathbf{H}$ can be formulated as

where $\mathbf{H}_{\mathrm{DL}}=\mathbf{U}_{2}\mathbf{H}^{T}\mathbf{T}_{2}$, $\tilde{\mathbf{x}}_{\mathrm{d},l}=\mathbf{W}_{\mathrm{t},l}\mathbf{x}_{\mathrm{d},l}$. Thanks to the proposed pilots and training scheme design, the above joint optimization problem can be equivalently decoupled into two subproblems demonstrated in the following proposition.

As $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are independent of each other, we first find the solution to $\mathcal{P}_{1}$, then solve $\mathcal{P}_{2}$. As the objective of $\mathcal{P}_{1}$ is bilinear, it can be solved by iterative approaches but this is inefficient. To solve $\mathcal{P}_{1}$ efficiently, we propose a closed-form solution by regarding the first receive digital chain of the UE as the calibration reference.

By using the auxiliary variables $\bar{\mathbf{X}}_{\mathrm{dr},p}=\mathbf{u}_{1}\mathbf{x}_{\mathrm{dt},p}^{T}$ and $\mathbf{x}_{\mathrm{dt},p}=\mathbf{X}_{\mathrm{dr}}^{T}\mathbf{t}_{1}$, the problem $\mathcal{P}_{1}$ can be further formulated by

By taking the derivative of the objective function of $\mathcal{P}_{1,1}$, the solution can be given by[36]

Since the first receive digital RF chain of the UE is the reference, its mismatch coefficient can be treated as a known constant, e.g., $u_{1,1}=c_{\mathrm{dr}}\neq 0$. Based on this assumption and (18), $\mathbf{x}_{\mathrm{dt},p}^{T}$ equals to the first column of $\bar{\mathbf{X}}_{\mathrm{dr},p}$, i.e.,

where $\mathbf{y}_{\mathrm{dr},m}$ is the $m$-th row of $\mathbf{Y}_{\mathrm{dr}}$.

By substituting (19) into (15), the solution to the problem $\mathcal{P}_{1}$ can be given in the following proposition.

Then, the mismatch coefficients of analog RF chains can be estimated by solving $\mathcal{P}_{2}$. By exploiting the geometry channel model of mmWave, the calibration problem of analog chains can be further written as

where $\tilde{\mathbf{X}}_{\mathrm{da}}=\bar{\mathbf{F}}_{\mathrm{da}}\mathbf{X}_{\mathrm{da}}$, $\mathbf{A}_{\mathrm{t}}=[\mathbf{a}_{\mathrm{t}}(\theta_{1}),\cdots,\mathbf{a}_{\mathrm{t}}(\theta_{K})]\in\mathbb{C}^{N_{\mathrm{t}}\times K}$, $\mathbf{A}_{\mathrm{r}}=[\mathbf{a}_{\mathrm{r}}(\phi_{1}),\cdots,\mathbf{a}_{\mathrm{r}}(\phi_{K})]\in\mathbb{C}^{N_{\mathrm{r}}\times K}$, and $\mathbf{H}_{\alpha}=\mathrm{diag}(\alpha_{1},\cdots,\alpha_{K})\sqrt{N_{\mathrm{t}}N_{\mathrm{r}}}/{\sqrt{K}}$. As the variables are correlated with each other, this problem is nonconvex and there is no tractable solution to the problem. To solve $\mathcal{P}_{2.1}$ efficiently, inspired by [39], we propose an alternating optimization algorithm to solve a locally optimal solution.

During the $l_{\mathrm{ao}}$-th iteration, we apply the least square algorithm to estimate the diagonal matrices $\mathbf{U}_{2}$, $\mathbf{T}_{2}$, $\mathbf{H}_{\alpha}$, then, propose an algorithm to estimate the AoA and AoD matrices $\boldsymbol{\Theta}$, $\boldsymbol{\Phi}$.

Finally, we propose a AoAs and AoDs updating algorithm¹¹1The update method of AoA/AoD is not restricted to the algorithm proposed in this paper. Some existing methods, such as those presented in [30], can be employed. when $\mathbf{U}_{2}^{l_{\mathrm{ao}}}$, $\mathbf{T}_{2}^{l_{\mathrm{ao}}}$, $\mathbf{H}_{\alpha}^{l_{\mathrm{ao}}}$, $\boldsymbol{\Theta}^{l_{\mathrm{ao}}-1}$, and $\boldsymbol{\Phi}^{l_{\mathrm{ao}}-1}$ are given. Since the AoAs and AoDs can be estimated using the same approaches, we introduce the updating algorithm by taking updating the AoDs $\boldsymbol{\Phi}$ as an example. Since the problem for estimating AoDs is nonlinear, it is difficult to solve the AoDs directly. To address this issue, as presented in [40], the nonlinear problem is transformed into a series of linear problems by using the first-order Taylor expansion to approximate the array steering vector. Let $\xi_{k}\ (k\in[1:K])$ denotes the differences between the estimated AoDs $\bar{\boldsymbol{\Phi}}$ and the real AoDs $\boldsymbol{\Phi}$. By assuming that the differences are small, the array steering vector can be approximated by the first-order Taylor expansion denoted as $\mathbf{a}_{\mathrm{r}}(\phi_{k})=\mathbf{a}_{\mathrm{r}}(\bar{\phi}_{k})+\bar{\mathbf{a}}_{\mathrm{r}}(\bar{\phi}_{k})\xi_{k}$, where $\bar{\mathbf{a}}_{\mathrm{r}}(\bar{\phi}_{k})=\partial\mathbf{a}_{\mathrm{r}}(\phi_{k})/\partial\phi_{k}|_{\phi_{k}=\bar{\phi}_{k}}=\mathbf{a}_{\mathrm{r}}(\bar{\phi}_{k})\circ[0,-j\frac{2\pi d}{\lambda}\cos\bar{\phi}_{k},\cdots,-j\frac{2\pi d}{\lambda}(N_{\mathrm{r}}-1)\cos\bar{\phi}_{k}]^{T}$. Then, the problem $\mathcal{P}_{2.1}$ can be further formulated as

where $\mathbf{H}_{\mathrm{r}}^{l_{\mathrm{ao}}}=\mathbf{H}_{\alpha}^{l_{\mathrm{ao}}}\mathbf{A}_{\mathrm{t}}^{T}\mathbf{T}_{2}^{l_{\mathrm{ao}}}\tilde{\mathbf{X}}_{\mathrm{da}}$, $\mathbf{Y}_{\mathrm{dar}}=\mathbf{Y}_{\mathrm{da}}-\bar{\mathbf{B}}_{\mathrm{da}}^{T}\mathbf{U}_{2}^{l_{\mathrm{ao}}}\mathbf{A}_{\mathrm{r}}(\bar{\boldsymbol{\Phi}})\mathbf{H}_{\mathrm{r}}^{l_{\mathrm{ao}}}$, $\boldsymbol{\Lambda}=\mathrm{diag}(\xi_{1},\cdots,\xi_{K})$, and $\bar{\mathbf{A}}_{\mathrm{r}}(\bar{\boldsymbol{\Phi}})=[\bar{\mathbf{a}}_{\mathrm{r}}(\bar{\phi}_{1}),\cdots,\bar{\mathbf{a}}_{\mathrm{r}}(\bar{\phi}_{K})]$. The solution can be given by